TABLE OF CONTENTS Various definitions of optimality of numerical methods and a detailed bibliography can be found in [7], [154]. Let us recall the definitions of algorithms, optimal with respect to accuracy, for calculating weakly singular integrals.
Consider the quadrature formula:
| (A.3) |  |
where coefficients 
and nodes 
are arbitrary. Here 
.
The error of quadrature formula (A.3) is defined as
The error of quadrature formula (A.3) on the class ? is defined as
Define the functional
The quadrature rformula with the coefficients 
and the nodes 
is optimal, asymptotically optimal, optimal with respect to order on the class ? among all quadrature rules of type (A.3) provided that:
The symbol ? ? ? means A ? ? ? ? B ?, where 0 < A, B < ?.
Consider the quadrature rule
| (A.4) |  |
where coefficients p k( t 1, t 2) and nodes ( M k) are arbitrary.
The error of quadrature formula (A.4) is defined as
The error of quadrature rule (A.4) on the class ? is defined as
Define the functional
The quadrature rule with the coefficients 
and the nodes 
is optimal, asymptotically optimal, optimal with respect to order on the class ? among all quadrature rules of the type (A.4) provided that:
| (A.5) |  |
By 
the error of optimal cubature formulas on the class ? is defined. One has 
.
